On a quantum-classical correspondence: from graphs to manifolds

We establish conditions for which graph Laplacians Δ λ , ϵ on compact, boundaryless, smooth submanifolds ℳ of Euclidean space are semiclassical pseudodifferential operators ( Ψ DOs). Specifically, the graph Laplacian’s kernel bandwidth ( bias term ) ϵ must decay faster than the semiclassical parameter h , i.e., h ≫ ϵ , and we compute the symbol. Coupling this with Egorov’s theorem and coherent states ψ h localized at ( x 0 , ξ 0 ) ∈ T * ℳ , we show that with U λ , ϵ t : = e - i t Δ λ , ϵ spectrally defined, the (co-)geodesic flow Γ t on T * ℳ is approximated by 〈 U λ , ϵ - t Op h ( a ) U λ , ϵ t ψ h , ψ h 〉 = a ∘ Γ t ( x 0 , ξ 0 ) + O ( h ) . In the discrete setting, for a normalized graph Laplacian Δ λ , ϵ , N defined on a set of N points x 1 , ⋯</